4  Population Mean Difference from Paired Data (8.2.7)

Sometimes the distributions that two samples come from are not independent, maybe because samples are taken from the same subjects at different points in time (e.g. test scores for a group of students before and after a revision lecture). When this is the case, the data is called paired data.

The observations in a paired sample of size \(n_D\) have values \(\left((x_1, y_1), (x_2, y_2),\dots , (x_{n_D}, y_{n_D}) \right)\), the difference between the measurements taken on the same subject is often of interest. These differences, calculated as \(D=(d_1=x_1-y_1, d_2=x_2-y_2,\dots ,d_{n_D}=x_{n_D}-y_{n_D})\), can be used to find a confidence interval for the mean difference.

Assuming a paired sample size of \(n_D\), a \((1-\alpha)\cdot100\%\) confidence interval for the difference in two dependent population means, \(\mu_X-\mu_Y\) is give by,

\[\begin{equation} \tag{8.21} CI_{1-\alpha}(\mu_D=\mu_X-\mu_Y)=\left[\bar d-t_{1-\frac{\alpha}{2};n_D-1}\cdot\frac{s_D}{\sqrt{n_D}},\,\,\bar d+t_{1-\frac{\alpha}{2};n_D-1}\cdot\frac{s_D}{\sqrt{n_D}}\right] \end{equation}\]

• \(\bar d\) is the mean of the sample differences \(d_1, d_2, \dots, d_{n_D}\).

• \(s_D\) is the standard deviation of the sampled differences. The variance of the sampled differences can be found using, \[s_D^2=\frac{\sum_{i=1}^{n_D}d_i^2-n\bar d^2}{n_D-1}\]

  Remember to take the square root of the variance to find the standard deviation.

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You can find some worked examples of using this result in Section 8.2.7 of Probability and Statistics with R.